Density of Cauchy initial data for solutions of elliptic equations

V. I. Voitinskii

Sbornik. Mathematics, Tome 14 (1971) no. 1, pp. 131-139

Résumé

In this paper we examine a problem connected with Cauchy's problem for linear elliptic equations. Let $G$ be a bounded region of $E_n$, and let $\Gamma$ be its boundary. In $G$ we consider the elliptic equation \begin{gather*} \mathscr Lu(x)=\sum_{|\mu|\leqslant 2m}a_\mu(x)D^\mu u(x)=0 \tag{1}\\ \biggl(\mu=(\mu_1,\dots,\mu_n);\quad|\mu|=\mu_1+\dots+\mu_n;\quad D^\mu=D_1^{\mu_1}\cdots D_n^{\mu_n},\quad D_k=-i\frac\partial{\partial x_k}\biggr), \end{gather*} where $\mathscr L$ is a regular elliptic expression with complex coefficients. Let $\Gamma_1$ be a piece of the surface $\Gamma$. The coefficients of the expression $\mathscr L$, the surface $\Gamma$, and the boundary $\Gamma_1$ are assumed to be infinitely smooth. We are concerned with Cauchy's problem on $\Gamma_1$ with the initial conditions $\{\partial^{j-1}u/\partial\nu^{j-1}|_{\Gamma_1}=f_j\}$, $j=1,\dots,2m$, where $\nu$ designates the direction normal to $\Gamma$. In this paper we prove that under our assumptions the set of Cauchy initial data for solutions of (1) in $H^l(G)$ is dense in $\sum_{j=1}^{2m}H^{l-j+1/2}(\Gamma_1)$ for any integer $l\geqslant2m$ if Cauchy's problem is unique for the formal conjugate operator $\mathscr L^+$, as is the case, for example, when $\mathscr L$ has no multiple complex characteristics. In addition, in this paper we give conditions under which the analogous assertion holds for certain elliptic systems. Bibliography: 4 titles.